Hi Arthur,

I hope I answer your question to some extent.

Consider the fixed-free beam column having length L loaded in compression with a compression load P at the free end. This theoretical beam column has all the ideal attributes (in reality not 100% achievable) as you suggested (like no eccentricity, no geometrical & material imperfections etc.).

At an arbitrary section at a distance x from the fixed end, by considering moment equilibrium we get equation:

M(x) = - ((F*x)/2 – P*v(x)

Where M(x) : Bending moment at the section x

            P : The applied compressive load

            F: The external destabilizing transverse load

           v(x): transverse displacement.

We now substitute this expression for  M(x) in to the Classical Beam Equation:                                                    d2v/dx2 = M(x)/ (EI)

Where E is elasticity and I the moment of inertia of the beam section.

Solving this equation for v(x), the transverse displacement due to destabilizing force F or buckling failure at a location x.  

This comes out to be:

v(x) = F*sin(Lamda*x)/(2EI*Lamda2*cos(Lamda*L/2)) – F*x/(2P)

Where Lamda = sqrt(P/EI)

When buckling failure is to occur, this elastic displacement v(x) will be maximum at x = L/2. So we’ll have elastic displacement v(x):

              v(L/2) = F/(2P)[ tan(Lamda*L)/2 – L/2]

              v(L/2) -----> infinity,  when (Lamda*L)/2 = pi/2

              v(L/2) -----> infinity (means irreversible instability leading to failure sets in)

Substituting this condition we get the critical value of load P at which the column just loses its elastic strength to restore back if it is made to have a transverse deformation. This Critical load Pcr comes out to be:

                Pcr = (pi2*E*I)/ L2

This is interesting and Arthur perhaps where your puzzle lies, at critical load there is no external force F. Which means when the column is loaded critically, You hardly need any external force to make the column buckle or fail. My Professor Lawrence so interestingly used to say “When the beam column is loaded critically, it will fail even if a butterfly flaps its wings somewhere in Africa”.

Hi Arthur,

Let me add my opinion to the conversation. The main concept to take into account is STABLE equilibrium vs UNSTABLE equilibrium. A good example is with a ball, please see the attached file (PDF) for a figure describing this.

Both states of the ball correspond to equilibrium, since F=W. In the first state (ball on valley) we have stable equilibrium, as small changes of the ball state would not lead to the ball falling. Small changes would be self-corrected and we would have equilibrium again.

In the second state of the ball (ball on top of hill), we have UNSTABLE equilibrium. Even very small changes on the state of the ball (very very small movement) would destroy the equilibrium and the ball would fall! 

The same is with a column. State a (stable) is when the column is in tension. Tension is a stable condition. State b (unstable) is when the column is in compression, close to the critical load.

In a "perfect" world, described by simple physics, state (b) is OK, as we have equilibrium and the ball does not fall. So ideally, there is no problem. The problem arises because nothing in the real world is perfect, so we have small imperfections. In states such as (a) (stable states), these small imprerfections are "corrected" by the system itself and there is no problem at all. In states such as (b) (unstable states), such small imperfections are not corrected - on the contrary, they are exaggerated and multiplied so the final result is failure.

In reality, we cannot achieve to build a perfect column or beam. Even if we use the most advanced techniques and the most advanced materials, it will not be perfect. In steel for example, one can see imperfections in the molecular connections and the bonds between molecules themselves. Nothing is perfect, even in the micro level and we cannot have control over these material properties. Let alone any small force caused by any source (for example slight wind) or even the fly of a butterfly in the region. That's why in reality the column will finally buckle!

I hope my reply helps!

Vagelis

KUMAR- your answer presupose that there'is a moment or eccentricity, which didn't actually happen!!! So, the perfect elastc hiperbolic solution will happen only if you can reach this load. But you will need some imperfection to start it break this equilibrium, walking away from the straight shape to the curved one. So, as to take you down the 'puzzle', let's give you more details: the column height is sufficiently low that elastic buckling load is about 5 times it's plastic load... (Pe > 5Py) so puzzle not solved!

PLEVRIS - that's what our puzzle is about. But, we must be more deeply engaged in the actual construction and material details. I already said that there's no such difference in the material properties it's self in the section or in the entire body of the actual column that we are analyzing. Therefore, there's something real to make such behavior. Why does precisetly this happen? How can I proof it in a mathematical model running on computer, for example? I know it's not easy, that's why I called it a puzzle. Puzzle not solved yet! 

It is well known (in mechanics of materials) that a concentrically loaded column will buckle when the applied load equal to or exceed Euler's critical buckling load. For eccentrically loaded columns, the buckling load will be smaller than that for the concentrically loaded one. 

ALTAAN_ sure, that's what I said before to others and that's the reason of my puzzle. Obviously, we are talking about no eccentricity of any type... at all. Puzzle not solved.

Somebody did upvoted some comentary I did, but I gave not the answer yet. I plan to give the answer near Christmas. The first "yes" was just given, the HOW, I'll give later as said. It's a verygood puzzle. Work on it!

Dear colegues,

maybe the answer of my puzzle is not too hard, if you look to my research' story.  Lets get into the question. If the column is said to be elastic-plastic, absolutely straight (mean no curvature nor out-of-plumb), no excentricity too and the load is applied absolutely centered, how can the column buckle? It can't, unless something make it behave differently when yielding begin. If it was elastic it would reach the squash load with no buckling.

But, we know that inside any steel section, there's residual stress, provided by the fabricating process (rolling, welding, etc). This stress has no resultant force, its auto equilibrated, but it appears since when steel is cooling after those process, out side parts will cool faster then the inner ones. As first parts cool, they contract, and the hot parts did follow due to its hot state.

So, when the still hot parts begin to cool too, its shrink is avoided by the just colded ones, which will not contract again. There will be a compression residual stress in the colded parts. Otherwise, the lastly colded parts are avoided to shrink and the tension stress appears there too.

Usually we researchers modeled the residual stresses symmetrically and identically on both flanges of  I sections. Now the puzzle comes. That's not necessarily true! If the rolled beam still hot is placed with one flange at the floor and other supporting other hot rolled beams, or one flange in the top of the pile, supported by the pile while the other flage is free (on the air), the cooling process will be different from one flange to another. That's the new!  Then you may conclude that one flange will have greater residual stress then the other, even  if their residual stress are equilibrated (no resultant).

So, when you apply the compression load, the flange with greater residual stress will reach the yielding first, this compression yielded zones will make the born of a eccentricity as there's no correspondence with what happens on the other side (that is elastic yet).  This will make the born of a bending moment, a horizontal displacement at the top of the column and a base rotation that make the member bucle inelastically. Amazzing? lgs.

This was published in the following paper. I hope you like the explanation.

Let me give you my latest Happy Christmas and new year everybody. Thanks to your time and appretiation. Sincerely yours,

ART.

Dear Arthur,

Thank you for your reply. I can say that I somehow disagree with your approach to this problem. You assume that steel is perfect until yield point and you accept that after yield there are symmetry (perfection) problems that cause buckling. Although nobody can disagree about the non-symmetry in the post-yield phase, the problem in your approach is that you assume perfection until yield which is not correct in reality and for any practical purpose.

In my opinion Steel is non-perfect before yield and also non-perfect after yield. Even molecules and bonds are not perfect! Only in theory is steel perfect and if it was perfect then it would not be able to buckle before yield. In reality, because of inherent imperfections even before yield, one can NEVER perform an experiment with a long and thin steel specimen and expect to compress it until yield. Far before yield, buckling occurs because of imperfections, even at the linear stage.

I don't disagree with your answer itself, I just add that the answer covers in detail the post-yielding explanation, when in reality there is similar behaviour even before yield, for other reasons due to imperfections!

Thank you for letting me share my opinion on this issue.

Vagelis

Dear Plevis,

of course steel is a non "perfect" material, you're right!, or best, it is like anything else we know in our real world, composed of a multitude of internal grains, of different shape, dimensions, posiitions, etc. There's no possibility ever that we have this wondered "perfect" material and steel is imperfect from the beginning.

But, when we study Stability, several problems are introduced by this wondered perfect material. And then something shake the equilibrium state, some imperfection comes in to start the buckling phenomena.

Imagine you could have some bubbles and inclusions inside some part of the column lenght, that you have no correpondence to the other parts of the section, which will induce bending and afterwards buckling. So I completely agree  with you.

But our goal is to show that even if that such thing could happen, if the steel is a perfect material (e.g. all cristals of the same type, dimension, distribution, no bubbles, no differences at all, perfect in all directions),  the presence of unequal residual stress could bring it to buckling. And this is not expected nor suspected.

I feel very happy for your answer, mean somebody spend  time on the puzzle. lgs. Thank you very much!

Dear Arthur,

I am glad that you understood my comment and received it the proper way with no offence. We certainly agree that steel is not perfect.

Thank you for highlighting this new aspect, that the presence of unequal residual stresses could lead to buckling. It is indeed very interesting and not expected!

Vagelis

Thanks Mr Arthur, for putting up a nice Puzzle and for its Ans.

Can I say, if there is no geometric or material imperfection, so called ideal column, will not fail in buckling ? Eulers buckling mode will not happen to this particular case.

You are wellcome Mr Nayak.

I think the main objective with my question it was to point the unexpected effect of the Residual Stress and the fact the we all suppose it is a symmetric thing, but it cannot be so.

The Euler buckling load was a very important contribution for the beggining of understanding the phenomenon. Later, among the laboratory tests, scientific and eletronic development, many other details appeared to create a new perspective about how it happens and how many subtle details are involved in the questions of Stability science.

The Residual Stress is only more one case. Surely, other will appear at the future. For example, maybe a compound material where the fibers are not symmetrically disposed or where for some particular reason, there appears a lateral deformation as the load is applied, which creates a non symmetric or out-of-balance component, which will be the start of the buckling.

In my Engineering Mechanics, its already know that the presence of some angular speed (rotation) of a circular column axis can lead to its buckling too.

The researcher must keep atention on for the multiples possibilities of imperfections and details that can lead some structure to unexpected disaster. We still continue to dream about this "perfect column" but surely we are certain to find it only in Paradise, where Euler will present it for us some day, laughting.

Sirs,

I have been pondering this exact question from my UG days, when I first learned about buckling. I had framed the question as :" how does symmetries (in force application, material homogeneity , cross section etc,) results in a unsymmetrical outcome, ie.the column bends to one particular direction. ?" If we never had seen buckling phenomenon before, we would have predicted the column to be squished uniformly in place, with outward bulging in all directions. But we know it buckles. Really weird, but understated aspect in engineering.

Now, I have two possible answers:

1. The stable equilibrium vs. Unstable equilibrium point. And we simply must accept that at unstable equilibrium ) a buckling load) , it an bend any direction arbitrarily. Here, we accept that instability is what matters and not any particular direction. The world around the column is inconsequential and the new bent state is only that matters. That is, the column doesn't care about the symmetry of the world.

Now, here's my favourite , but outlandish explanation:

2. Assuming perfect symmetry n material, we reach into and into the material til the quantum level. Now, the combined wave function of a particular section is a probability function, mostly centred around the centre of section. But when critical load is reached, the wave function gets stretched out more and more until a high probability (newly) exists for the average to not be coinciding with the section centre. Now, you may argue that how come it only goes to one particular direction and not uniformly in all. That I believe is the unpredictability of quantum mechanics, where the probability of a particles position can vary about a point , like a cloud, unpredictability.

Basically what I'm arguing is that bucking is the macro world realisation of a quantum phenomenon.

I'm stil working on this matter. So see you.

Regards

Siddharth George

Hi again.

Just a quick clarification regarding the quantum explanation.

Below critical load, the average of wave fn is at centre , effectively. Than at critical load, the wave fn crowds at a point/points( maybe even on a circle with the centre at centre of section). This the fn has equal chances of being at any point on this new circle (area), which leads to unsymmetry in material and subsequent buckling along that direction.

Thanks